Hypermatrix factors for string and membrane junctions

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n) and Sp(n) are, respectively, tensor, antisymmetric and symmetric products of two vector spaces, and hence are matrix representations. We consider the analogous products of three vector spaces and study when they appear as summands in Lie algebra decompositions. The Z3-grading of the exceptional Lie algebras provides such summands and provides representations of classical groups on hypermatrices. The main natural application is a formal study of 3-junctions of strings and membranes. Generalizations are also considered.